Find the first partial derivatives of the function. f(x,t)=xln⁡t

Question

Find the first partial derivatives of the function.  f(x,t)=xln⁡t

Dona Hall · Accepted Answer

We will first find the partial with respect to x and treat t as a constant

f_{x} = \frac{1}{2 \sqrt{x}} \ln t

Now we will find the partial derivative of f with respect to t, treating x as a constant:

f_{t} = \sqrt{x} \frac{1}{t}

Result:

f_{x} = \frac{\ln t}{2 \sqrt{x}}

f_{t} = \frac{\sqrt{x}}{t}

madeleinejames20 · Accepted Answer

Result:∂f∂x=0and∂f∂t=0Solution:Let&#039;s start by finding the partial derivative with respect to x, denoted as ∂f∂x.∂f∂x=∂∂x(xlnt)To differentiate x, we can use the power rule for differentiation. Since x=x12, we have:∂∂x(x)=12x−12Next, we differentiate lnt with respect to x. Since t does not contain x, its derivative with respect to x is zero:∂∂x(lnt)=0Now we can find the first partial derivative of f with respect to x by multiplying the derivatives of the individual terms:∂f∂x=(12x−12)·0=0Therefore, the first partial derivative of f(x,t) with respect to x is 0.Next, let&#039;s find the partial derivative with respect to t, denoted as ∂f∂t.∂f∂t=∂∂t(xlnt)To differentiate x with respect to t, we treat it as a constant since x does not contain t:∂∂t(x)=0Next, we differentiate lnt using the chain rule. The derivative of lnt with respect to t is 1t:∂∂t(lnt)=1tMultiplying the derivatives of the individual terms, we find:∂f∂t=0·1t=0Therefore, the first partial derivative of f(x,t) with respect to t is 0.In conclusion, the first partial derivatives of the function f(x,t)=xlnt are:∂f∂x=0and∂f∂t=0

Nick Camelot · Accepted Answer

The first partial derivatives of the function f(x,t)=xlnt can be found as follows:∂f∂x=12xlnt∂f∂t=xt

Mr Solver · Accepted Answer

Step 1: First, we&#039;ll find the partial derivative with respect to x (∂f∂x):∂f∂x=∂∂x(xlnt)Using the product rule, we differentiate x and lnt separately:∂∂x(xlnt)=12xlntSo, the first partial derivative with respect to x is:∂f∂x=12xlntStep 2: Now, let&#039;s find the partial derivative with respect to t (∂f∂t):∂f∂t=∂∂t(xlnt)Here, we differentiate lnt while treating x as a constant:∂∂t(xlnt)=x·1t=xtTherefore, the first partial derivative with respect to t is:∂f∂t=xtStep 3: To summarize, the first partial derivatives of the function f(x,t)=xlnt are:∂f∂x=12xlnt∂f∂t=xt

Find the first partial derivatives of the function. f(x, t)=\sqrt{x}\ln t

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